Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10,15,20)(11,16,22,24)(12,13,18,17)(14,23,21,19)(25,26,30,32)(27,31,36,40)(28,29,37,39)(33,34,35,38);(1,3,6,5)(2,7,4,8)(9,11,15,22)(10,16,20,24)(12,21,18,14)(13,19,17,23)(25,27)(26,31)(28,35)(29,38)(30,36)(32,40)(33,37)(34,39);(9,12)(10,17)(11,21)(13,20)(14,22)(15,18)(16,23)(19,24)(25,28)(26,29)(27,33)(30,37)(31,34)(32,39)(35,36)(38,40);(1,2,6,4)(3,8,5,7)(9,12)(10,13)(11,21)(14,22)(15,18)(16,19)(17,20)(23,24)(25,29)(26,28)(27,34)(30,39)(31,33)(32,37)(35,40)(36,38);(1,4,6,2)(3,8,5,7)(9,13)(10,18)(11,19)(12,20)(14,16)(15,17)(21,24)(22,23)(25,28)(26,29)(27,35)(30,37)(31,38)(32,39)(33,36)(34,40);(1,5,6,3)(2,7,4,8)(9,14,15,21)(10,19,20,23)(11,12,22,18)(13,16,17,24)(25,27)(26,31)(28,33)(29,34)(30,36)(32,40)(35,37)(38,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10365124);